Given-the-function-f-x-6x-2-x-3-1-3-Find-the-oblique-assymptote-s-of-the-function-

Question Number 145286 by physicstutes last updated on 03/Jul/21

Given the function

f (x) = \sqrt[3]{6 x^{2} - x^{3}}

Find the oblique assymptote (s) of the function .

Answered by Olaf_Thorendsen last updated on 04/Jul/21

f (x) = \sqrt[3]{6 x^{2} - x^{3}}

f (x) = - x \sqrt[3]{1 - \frac{6}{x}}

f (x) \underset{\infty}{\sim} - x (1 + \frac{1}{3} (- \frac{6}{x}))

f (x) \underset{\infty}{\sim} - x + 2

Commented by physicstutes last updated on 04/Jul/21

Thanks sir but my textbook came up with this argument which

makes me confused .

f (x) = \sqrt[3]{6 x^{2} - x^{3}}

f (x) = ∣ x ∣ \sqrt[3]{\frac{6}{x} - 1}

as x \to \pm \infty, f (x) \to - x hence y = - x is an oblique

asymptote .

How true is that sirs ?

Answered by mathmax by abdo last updated on 04/Jul/21

\lim_{x \to \infty} \frac{f (x)}{x} = \lim_{x \to \infty}^{3} \sqrt{\frac{- x^{3} + {6 x}^{2}}{x^{3}}} = \lim_{x \to \infty} \sqrt{- 1 + \frac{6}{x}} = - 1

\lim_{x \to \infty} f (x) + x = \lim_{x \to \infty} x +^{3} \sqrt{{6 x}^{2} - x^{3}}

we have {(- x^{3} + {6 x}^{2})}^{\frac{1}{3}} = - x {(1 - \frac{6}{x})}^{\frac{1}{3}} \sim - x (1 - \frac{2}{x}) = - x + 2 \Rightarrow

f (x) + x \sim 2 \Rightarrow y = - x + 2 is assymptote oblique for C_{f}

Answered by imjagoll last updated on 04/Jul/21

let y = mx + c oblique asymptotes

(1) m = \lim_{x \to \infty} \frac{y}{x} = \lim_{x \to \infty} \frac{\sqrt[3]{{6 x}^{2} - x^{3}}}{x}

= \lim_{x \to \infty} \frac{\sqrt[3]{{6 x}^{2} - x^{3}}}{\sqrt[3]{x^{3}}} = - 1

(2) c = \lim_{x \to - \infty} (y - mx)

c = \lim_{x \to - \infty} (\sqrt[3]{{6 x}^{2} - x^{3}} + x

c = \lim_{x \to - \infty} - x (\sqrt[3]{- {6 x}^{- 1} + 1} - 1)

c = \lim_{t \to 0} \frac{\sqrt[3]{6 t + 1} - 1}{t}, [- x = \frac{1}{t}]

c = \lim_{t \to 0} \frac{6 t}{t} \times \lim_{t \to 0} \frac{1}{\sqrt[3]{{(6 t + 1)}^{2}} + 1 + \sqrt[3]{6 t + 1}}

c = 6 \times \frac{1}{3} = 2

thus oblique asymptotes is

y = - x + 2

Commented by imjagoll last updated on 04/Jul/21